The jiggle that proved atoms

Robert Brown, a botanist, looked at pollen grains suspended in water in 1827 and found them in constant, irregular motion that never stopped. He suspected life, tested the idea by grinding up material that had never been alive — including a fragment of the Sphinx — and found the same motion. Then he published the observation and no explanation, which was the correct thing to do.

The explanation took until 1905, and when it arrived it was not primarily an explanation. It was a measurement proposal, and it settled an argument that had been running for a century about whether atoms exist.

What is doing the hitting #

A particle a micron across, suspended in water, is being struck by water molecules from all sides. The number of impacts is enormous — of order $10^{20}$ per second — and if they were perfectly balanced the particle would sit still.

They are not perfectly balanced. Over any short interval the number arriving from the left differs from the number arriving from the right by roughly $\sqrt{N}$ out of $N$, which is the ordinary statistical fluctuation of a random count. With $N \sim 10^{20}$, the fractional imbalance is about $10^{-10}$ — utterly negligible as a force at any instant.

The reason it is nonetheless visible is that the imbalances do not cancel over time. They are random and independent, so they accumulate, and what accumulates is not the displacement but its square. That distinction is the whole of the quantitative theory.

Why the square of the distance is the useful quantity #

Model the motion as a walk: a sequence of independent steps, each of length $\ell$, each in a random direction. After $n$ steps the displacement is the vector sum

$$\mathbf{R} = \sum_{i=1}^{n} \boldsymbol{\ell}_i.$$

Its average is zero — every direction is as likely as its opposite, so the particle has no preferred destination. That is why the displacement itself carries no information: measure enough particles and the mean is nothing.

Square it instead:

$$\langle R^2 \rangle = \sum_i \langle \ell_i^2 \rangle + \sum_{i \neq j} \langle \boldsymbol{\ell}_i \cdot \boldsymbol{\ell}_j \rangle = n\ell^2 + 0.$$

The cross terms vanish because successive steps are uncorrelated, and what survives is $n\ell^2$: the mean square displacement grows in proportion to the number of steps, and therefore in proportion to time.

So the typical distance grows as $\sqrt{t}$. Not as $t$ — that would be a drift, a particle going somewhere. The difference between the two is the difference between being pushed and being jostled, and it is directly measurable.

Einstein’s relation #

Einstein’s 1905 paper did two things. It identified the wandering with ordinary diffusion — so that a cloud of such particles spreads by the same equation as a drop of ink — and it computed the diffusion coefficient from measurable quantities.

The argument balances two effects. A concentration gradient of suspended particles drives a diffusive flux. Gravity, or any applied force, drives them the other way, at a drift speed set by the viscous drag on a sphere. In equilibrium the two must cancel, and the equilibrium distribution is fixed by the Boltzmann factor. Requiring consistency gives

$$D = \frac{k_B T}{6\pi\mu a},$$

with $a$ the particle’s radius and $\mu$ the fluid’s viscosity, and hence

$$\langle x^2 \rangle = 2Dt.$$

The structure of that relation is worth pausing on. On the left is a fluctuation — the random wandering. On the right is a dissipation — the viscous drag, which is what converts organised motion into heat. The two are tied together, and they must be, because both come from the same molecular bombardment: the impacts that push a stationary particle around are the impacts that resist a moving one.

This is the first instance of what became the fluctuation–dissipation theorem, and it recurs everywhere. Johnson noise in a resistor is the same statement, with electrical resistance in place of viscosity — and the same statistical reasoning about counts underlies both.

Why the obvious measurement fails #

Before the useful measurement, it is worth seeing why the obvious one is useless — because the failure is instructive and it delayed the subject for decades.

The natural thing to do with a jiggling particle is to measure its speed: track it, divide distance by time, and compare with what equipartition predicts. Equipartition gives $\tfrac12 m \langle v^2 \rangle = \tfrac32 k_B T$, and for a micron particle that is a few millimetres per second — easily fast enough to measure.

The measurement gives nothing of the sort. Track the same particle at a finer time resolution and the apparent speed comes out larger; go finer still and it grows again. There is no converged answer, because the path is not differentiable: at every scale it is a jagged walk, and distance divided by time diverges as the time interval shrinks.

Einstein’s contribution was partly to see that this is not a defect of the measurement but a property of the motion, and that the quantity to measure is therefore not a velocity but a displacement over a stated interval. The theory predicts $\langle x^2 \rangle = 2Dt$ and says nothing at all about instantaneous speed, which is exactly why it is testable.

That is a general lesson worth carrying. A quantity that seems obviously measurable can fail to exist, and the theory’s real content is often the statement of which quantity to look at — a point the collection makes elsewhere about the pointwise limit of a probability density, where the naive quantity also fails to converge and a windowed one does.

The count of arrangements peaks ever more sharply as a system grows, so a fractional fluctuation falls as one over the square root of the number of particles. That is why the obvious measurement fails: a visible object contains so many molecules that its fluctuations are unmeasurably small, and the effect is only visible on something small enough to have a modest number of collisions and large enough to be seen. That window is a micrometre wide, which is why the discovery waited for a microscope.

The measurement that ended the argument #

The point of the formula is that everything in it except $k_B$ is measurable with a microscope and a thermometer. Watch a particle of known radius in a fluid of known viscosity at known temperature, record its position at regular intervals, compute the mean square displacement, and $k_B$ falls out. And since the gas constant $R$ was known, $N_A = R/k_B$ follows: the number of molecules in a mole, from watching a speck of resin through a microscope.

Jean Perrin did exactly this between 1908 and 1911, with painstakingly size-sorted gamboge particles. He got $N_A \approx 6.5 \times 10^{23}$ against the modern $6.022 \times 10^{23}$ — a few per cent, from position measurements made by hand.

What made it decisive was not the accuracy but the agreement across methods. Perrin obtained $N_A$ several independent ways — from the mean square displacement, from the rotational wandering, from the vertical distribution of particles in gravity — and cross-checked against values from viscosity of gases, from blackbody radiation, from radioactivity. They agreed.

That convergence is what ended serious opposition to the atomic hypothesis. Ostwald, who had argued for years that atoms were a useful fiction, accepted them in 1909 and said why: it was the Brownian evidence. A hypothesis that had been philosophically contentious for a century was settled by a number obtained six ways.

Suspended particles settle into an exponential distribution with height, exactly as the atmosphere does — with the scale height set by the particle’s mass instead of a molecule’s. Perrin measured that distribution and got the same Avogadro number he got from the wandering, which is the part that ended the argument: two quite different measurements on the same suspension agreeing to within a few per cent.

The scale that makes it visible #

Brownian motion is a competition between thermal energy and size, and the reason it is a microscope phenomenon rather than an everyday one is worth quantifying.

$D$ goes as $1/a$, so larger particles diffuse more slowly. For a one-micron sphere in water, $D \approx 4 \times 10^{-13}$ m²/s, giving about 0.9 microns of wandering per second — clearly visible under a microscope. For a one-millimetre sphere, $D$ is a thousand times smaller and the displacement in a second is about 30 nanometres, which is invisible. For a football, it is unmeasurable by any means.

At the other end it grows: a protein a few nanometres across diffuses at around $10^{-10}$ m²/s and crosses a bacterium in a fraction of a second. Molecular transport inside a cell is largely diffusion, and it works because the distances are small — which is exactly the square-root law being unforgiving over long distances and generous over short ones.

That scaling explains the size of cells. Diffusion crosses a micron in about a millisecond and a metre in about thirty years, so anything larger than a fraction of a millimetre needs active transport or circulation. Being small enough to rely on diffusion is a design constraint, and everything above that size has a plumbing system. That is the same square-root arithmetic that makes stirring worthwhile: folding a fluid shortens the distances diffusion has to cross, and because the time goes as the distance squared, halving the distance quarters the wait.

Histogram many walkers at once against the exact solution of the diffusion equation and the two agree. That is the bridge the whole subject needed: the individual wandering is unpredictable and the population’s spreading is smooth and exactly calculable, so a molecular hypothesis about individuals produces a continuum prediction that can be tested without ever following one particle.

What it says about the second law #

There is a consequence that troubled people at the time and is worth stating plainly.

A Brownian particle spontaneously moves. Sometimes it moves upward against gravity; sometimes it happens to receive a run of impacts from one side and travels a visible distance in one direction. In those moments the particle’s energy has increased at the expense of the surrounding liquid’s thermal energy, which is exactly what the second law is usually said to forbid.

The resolution is that the second law is a statement about probabilities, and it becomes a certainty only for large numbers. A Brownian particle is small enough that fluctuations are visible, so it violates the naive statement routinely, briefly, and by tiny amounts — and never in a way that can be accumulated into useful work, because any mechanism that tried to rectify the fluctuations would itself be subject to them.

This is not a loophole; it is what the statistical foundation implies. Brownian motion is the second law’s fine print made visible, and it is the most direct evidence available that thermodynamics is statistics rather than law.

The same wandering, put to work #

Once the relation is trusted, it can be run in either direction, and both are routine.

Sizing particles. Measure the diffusion coefficient of a suspension by watching how fast scattered laser light fluctuates, invert $D = k_BT/6\pi\mu a$, and out comes the particle radius. Dynamic light scattering does this on samples of a few microlitres and is the standard way of sizing nanoparticles, protein aggregates and emulsion droplets. What is being measured is Brownian motion, at a rate too fast to watch.

Measuring viscosity. With a particle of known size, the same relation gives $\mu$ — and it does so using a nanolitre of sample and no moving parts, which is why microrheology exists. It also gives the viscosity locally, inside a cell or a gel, where no conventional instrument fits.

Weighing single molecules. A fluorescent molecule’s diffusion coefficient depends on its size, so watching one wander through a focused laser spot reveals whether it has bound to a partner. Fluorescence correlation spectroscopy is built on this and detects binding one molecule at a time.

Setting a limit. In any small sensor — a micro-cantilever, an optical trap, a MEMS gyroscope — the thermal wandering of the moving part is a noise floor that no improvement in electronics can lower. It is set by $k_BT$ and by the dissipation, and the only ways down are to make the device stiffer or to cool it.

That last one is worth dwelling on. The same relation that made Brownian motion a proof of atoms now sets the sensitivity limit of a great deal of precision instrumentation — and it does so because fluctuation and dissipation cannot be separated. A detector coupled strongly enough to its surroundings to be read out is coupled strongly enough to be kicked.

The molecules doing the kicking have a distribution of speeds, and that distribution is what makes the bombardment statistical rather than steady. It is also where the temperature enters: the diffusion coefficient carries $kT$, so measuring how far a particle wanders in a known time and at a known temperature yields Boltzmann’s constant — and dividing the gas constant by it yields Avogadro’s number.

What Brown actually deserves credit for #

It is worth correcting the usual telling, because Brown is often described as having observed something and failed to explain it, which understates him considerably.

Motion of suspended particles had been noticed before him. What Brown did was to establish, by a systematic series of controls, that it was not what everybody assumed. He began with pollen and the hypothesis that it was vital motion — the grains were, after all, reproductive material. He then repeated the observation with pollen from herbarium specimens dead for a century, with ground glass, with soot, with metals, and with powdered rock from the Sphinx.

Every one of them showed the same motion. He also ruled out evaporation currents, convection from illumination and the effect of the observer’s own movement, by sealing drops in oil and by varying every condition he could. What he published was therefore not a curiosity but a negative result of considerable strength: whatever this is, it is a property of small particles in fluids and not of life, chemistry or the apparatus.

That is precisely the shape of contribution this collection keeps finding valuable — an assertion given a test it could fail, several times over, in different materials. The explanation was eighty years away; the elimination of the wrong ones was complete in 1827, and it is why nobody had to re-do it.

A light fast molecule striking a much heavier particle transfers a small momentum, and the visible wandering is the accumulated effect of an enormous number of such events. That is what Brown actually deserves credit for: not explaining the motion, which he did not, but establishing that it was not biological — he saw it in dust from a fragment of the Sphinx, which had not been alive for some time.

The noise that calibrates the instrument #

The most elegant modern use of this relation is one in which the jiggling is not the signal and not the noise, but the ruler.

Hold a micron bead in a focused laser beam and it sits in a potential well: displace it and the light pushes it back, with a restoring force proportional to the displacement for small excursions. The bead is now a Brownian particle in a spring, and equipartition says that its mean square displacement is $k_BT$ divided by the spring’s stiffness.

So watching the bead rattle measures the stiffness. Record its position for a second, take the variance, divide $k_BT$ by it, and the trap is calibrated — in newtons per metre, absolutely, with no reference standard and nothing to compare against. A trap of a tenth of a piconewton per nanometre gives a bead an excursion of about six nanometres, which a quadrant detector reads easily.

Once the stiffness is known the bead is a force meter, because the displacement times the stiffness is the force being applied to it. Attach the bead to a single molecule of a motor protein walking along a filament, and the trap reports the force the molecule exerts and the distance it moves — which is how the eight-nanometre steps of kinesin, and the few piconewtons at which it stalls, were measured one molecule at a time.

The pleasing part is the circularity. Thermal fluctuation is what makes the measurement noisy, and thermal fluctuation is what makes the measurement absolute. There is no way to have one without the other, and the instrument’s calibration is its own noise floor read as a signal.

The wandering that is far more sensitive to size #

Everything above is about a particle wandering in position. It also wanders in orientation, and the rotational version has a scaling that makes it a much sharper probe.

The rotational diffusion coefficient of a sphere is $k_BT/8\pi\mu a^3$ — the cube of the radius rather than the first power. So doubling a particle’s size halves its translational diffusion and slows its tumbling by a factor of eight.

For a protein a couple of nanometres across in water that gives a tumbling time of a few nanoseconds, which is conveniently comparable with how long a fluorescent dye stays excited. Excite such a molecule with polarised light and it emits polarised light too, unless it has turned appreciably in the meantime — so the polarisation of the emission reports how fast the molecule is tumbling, and therefore how large it is.

Because of the cube, that is a discriminating measurement. A small labelled molecule binding to a large partner increases its volume many times over and slows its tumbling in proportion, taking the emission from nearly unpolarised to strongly polarised. Fluorescence anisotropy assays are built on exactly that, and they detect a binding event as a change in the polarisation of a light beam.

Perrin measured rotational Brownian motion too, on visible particles under a microscope, as one of his independent routes to Avogadro’s number. The same quantity is now a laboratory workhorse, at a scale a thousand times smaller and a timescale a billion times shorter.

Where the model stops #

Steps are independent. They are not, on short enough timescales. A real particle has inertia and its velocity persists for a time of order $m/6\pi\mu a$ — about 0.1 microseconds for a micron sphere in water. Below that the motion is smooth and ballistic rather than diffusive, and $\langle x^2 \rangle \propto t^2$. That regime was thought unobservable and was measured directly in 2010.

The fluid is a continuum. The derivation uses a viscosity and a drag law, which presuppose many molecules per particle diameter. For a molecule diffusing among molecules of similar size, the Stokes drag is not right and the relation becomes an approximation.

The particle is a sphere and the fluid unbounded. Near a wall the drag rises and the diffusion slows, which matters throughout microfluidics and in every measurement done in a thin cell.

No interactions between particles. At higher concentrations they interact hydrodynamically and the collective diffusion differs from the single-particle one.

The bombardment is uncorrelated in time. The molecules hitting the particle are themselves interacting, and on timescales comparable with the interval between molecular collisions the impacts are not independent. That is far below anything observable for a micron particle and it is the reason the step model works at all.

Equilibrium. Living systems are full of motion that looks Brownian and is not, being driven by molecular motors consuming energy. Distinguishing the two requires checking whether the fluctuation–dissipation relation actually holds, which is a real experimental subject and which turns the relation on this page into a test for whether something is alive rather than merely warm.

The ladder from here #

Later rungs on this anchor: the Langevin equation, which puts a random force into Newton’s second law and recovers all of this. The fluctuation–dissipation theorem in general. Johnson noise as the electrical case. The ballistic regime and its measurement. Diffusion in crowded and confined environments, where the exponent departs from one. And the Brownian ratchet, where the impossibility of rectifying thermal fluctuations is made precise — and where the ways real molecular motors evade the argument turn out to be instructive rather than exceptions.